REDUCTION OF DERIVED HOCHSCHILD FUNCTORS OVER COMMUTATIVE ALGEBRAS AND SCHEMES

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1 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS OVER COMMUTATIVE ALGEBRAS AND SCHEMES LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, JOSEPH LIPMAN, AND SURESH NAYAK Abstract. We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext S L K SS K; M L K N) Ext S RHom SM, D), N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology HM) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite-type flat map f : X Y of noetherian schemes, with f! O Y in place of D. Contents Introduction 1 1. Relative dualizing complexes 3 2. DG derived categories 8 3. Derived Hochschild functors Reduction of derived Hochschild functors over algebras Global duality Reduction of derived Hochschild functors over schemes 27 Acknowledgments 31 References 31 Introduction We study commutative algebras essentially of finite type over some commutative noetherian ring K. Let σ : K S denote the structure map of such an algebra. When S is projective as a K-module, for example, when K is a field, the Hochschild cohomology HH S K; ) allows one to investigate certain properties of the homomorphism σ in terms of properties of S, viewed as a module over the enveloping Date: January 9, Mathematics Subject Classification. Primary 13D03, 14B25. Secondary 14M05, 16E40. Key words and phrases. Hochschild derived functors, Hochschild cohomology, homomorphism essentially of finite type, smooth homomorphism, relative dualizing complex, Grothendieck duality. Research partly supported by NSF grants DMS and DMS LLA), DMS SBI), and NSA grant H JL). 1

2 2 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK algebra S e = S K S. This comes about via isomorphisms HH n S K; L) = Ext n SeS, L), established by Cartan and Eilenberg [10] for an arbitrary S-bimodule L. In the absence of projectivity, one can turn to a cohomology theory introduced by MacLane [21] for K = Z, extended by Shukla [28] to all rings K, and recognized by Quillen [26] as a derived version of Hochschild cohomology; see Section 3. A central result of this article is a reduction of the computation of derived Hochschild cohomology with coefficients in M L K N to a computation of iterated derived functors over the ring S itself; this is new even in the classical situation. We write DS) for the derived category of S-modules, and Pσ) for its full subcategory consisting of complexes with finite homology that are isomorphic in DK) to bounded complexes of flat K-modules. As part of Theorem 4.1 we prove: Theorem 1. When S has finite flat dimension as a K-module there exists a unique up to isomorphism complex D σ Pσ), such that for each M Pσ) and every N DS) there is an isomorphism that is natural in M and N: RHom S L K SS, M L K N) RHom S RHom S M, D σ ), N). The complex D σ is an algebraic version of a relative dualizing complex used in algebraic geometry, see 6.2.1). A direct, explicit construction of D σ is given in Section 1. When S is flat as a K-module, M and N are S-modules, and M is flat over K and finite over S, the theorem yields isomorphisms of S-modules Ext n S es, M K N) = Ext n SRHom S M, D σ ), N) for all n Z; they were originally proved in the first preprint version of [5]. Our second main result is a global version of part of Theorem 1. For a map of schemes f : X Y, f0 1 O Y is a sheaf of commutative rings on X, whose stalk at any point x X is O Y,fx) see Section 6). The derived category of sheaves of) f0 1 O Y -modules is denoted by Df0 1 O Y ). Corollary 6.5 of Theorem 6.1 gives: Theorem 2. Let f : X Y be an essentially finite-type, flat map of noetherian schemes; let X π1 X Y X π2 X be the canonical projections; let δ : X X Y X be the diagonal morphism; and let M and N be complexes of O X -modules. If M has coherent cohomology and is isomorphic in Df0 1 O Y ) to a bounded complex of f0 1 O Y -modules that are flat over Y, and if N has bounded-above quasicoherent homology, then one has an isomorphism δ! π 1M L X Y X π 2N) RHom X RHom X M, f! O Y ), N). When both schemes X and Y are affine, and f corresponds to an essentially finite-type ring homomorphism, Theorem 2 reduces to a special case of Theorem 1, namely, where the K-algebra S is flat and N is homologically bounded above. In Section 6 we also obtain global analogs of other results proved earlier in the paper for complexes over rings. A pattern emerging from these series of parallel results is that neither version of a theorem implies the other one in full generality. This intriguing discrepancy suggests the existence of stronger global results. The proofs of Theorems 1 and 2 follow very different routes. The first one is based on isomorphisms in derived categories of differential graded algebras; background material on the topic is collected in Section 2. The second one involves fundamental results of Grothendieck duality theory, systematically developed in [15, 11, 19]; the relevant notions and theorems are reviewed in Section 5.

3 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 3 1. Relative dualizing complexes In this section σ : K S denotes a homomorphism of commutative rings. For any K-algebra P and each n Z we write Ω P K for the P -module of Kähler differentials of P over K, and set Ω n P K = n P Ω P K for each n N. Recall that σ is said to be essentially of finite type if it can be factored as 1.0.1) K K[x 1,..., x e ] V 1 K[x 1,..., x e ] = Q S, where x 1,..., x e are indeterminates, V is a multiplicatively closed set, the first two maps are canonical, the equality defines Q, and the last arrow is a surjective ring homomorphism. We fix such a factorization and set 1.0.2) D σ = Σ e RHom P S, Ω e Q K ) in DS), where DS) denotes the derived category of S-modules. Any complex isomorphic to D σ in DS) is called a relative dualizing complex of σ. To obtain such complexes we factor σ through essentially smooth maps, see 1.3. Theorem 1.1. If K P S is a factorization of σ, with K P essentially smooth of relative dimension d and P S finite, then there exists an isomorphism D σ Σ d RHom P S, Ω d P K ) in DS). The isomorphism in the theorem can be chosen in a coherent way for all K- algebras essentially of finite type. To prove this statement, or even to make it precise, we need to appeal to the theory of the pseudofunctor! of Grothendieck duality theory; see [19, Ch. 4]. Canonicity is not used in this paper. We write Pσ) for the full subcategory of DS) consisting of complexes M DS) such that HM) is finite over S and M is isomorphic in DK) to some bounded complex of flat K-modules. The name given to the complex D σ is explained by the next result. Theorem 1.2. When fd K S is finite the complex D σ has the following properties. 1) For each M in Pσ) the complex RHom S M, D σ ) is in Pσ), and the biduality morphism gives a canonical isomorphism δ M : M RHom S RHom S M, D σ ), D σ ) in DS). 2) One has D σ Pσ), and the homothety map gives a canonical isomorphism χ Dσ : S RHom S D σ, D σ ) in DS). The theorems are proved at the end of the section. The arguments use various properties of essentially) smooth homomorphisms, which we record next Let κ : K P be a homomorphism of commutative noetherian rings. One says that κ : K P is essentially) smooth if it is essentially) of finite type, flat, and the ring k K P is regular for each homomorphism of rings K k when k is a field; see [14, ] for a proof that this notion of smoothness is equivalent to that defined in terms of lifting of homomorphisms. When κ is essentially smooth Ω 1 P K is finite projective, so for each prime ideal p of P the P p -module Ω 1 P K ) p is free of finite rank. If this rank is equal to a fixed integer d for all p, then K P is said to be of relative dimension d; essentially) smooth homomorphism of relative dimension zero are called essentially) étale.

4 4 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK Set P e = P K P and I = Kerµ: P e P ), where µ is the multiplication. There exist canonical isomorphisms of P -modules Ω 1 P K = I/I 2 = Tor P e 1 P, P ). As µ is a homomorphism of commutative rings, Tor P e P, P ) has a natural structure of a strictly graded-commutative P -algebra, so the composed isomorphism above extends to a homomorphism of graded P -algebras λ P K : P Ω1 P K TorP e P, P ) Let X P be a projective resolution over P e. The morphism of complexes δ : X P e P Hom P ehom P ex, P e ), P ) δx p)χ) = 1) x + p ) χ χx)p yields the first map in the composition below, where κ is a Künneth homomorphism: HX P e P ) Hδ) HHom P ehom P ex, P e ), P )) κ Hom P ehhom P ex, P e )), P ) Hom P HHom P ex, P e )), P ). Thus, one gets a homomorphism of graded P -modules τ P K : Tor P e P, P ) Hom P Ext P ep, P e ), P ) The composition below, where the first arrow is a biduality map, Ext P ep, P e ) Hom P Hom P Ext P ep, P e ), P ), P ) is a homomorphism of graded P -modules Hom P τ P K,P ) Hom P Tor P e P, P ), P ). ɛ P K : Ext P ep, P e ) Hom P Tor P e P, P e ), P ). The maps above appear in homological characterizations of smoothness: Let K P be a flat and essentially of finite type homomorphism of rings, and set I = Kerµ: P e P ). The following conditions are equivalent. i) The homomorphism K P is essentially smooth. ii) The ideal I m is generated by a regular sequence for each prime ideal m I. iii) The P -module Ω 1 P K is projective and the map λp K from is bijective. iv) The projective dimension pd P e P is finite. The equivalence of the first three conditions is due to Hochschild, Kostant, and Rosenberg when K is a perfect field, and to André [1, Prop. C] in general. The implication ii) = iv) is clear, and the converse is proved by Rodicio [27, Cor. 2]. In the next lemma we use homological dimensions for complexes, as introduced in [3]. They are based on notions of semiprojective and semiflat resolutions, recalled in The projective dimension of M DP ) in defined by the formula { } n sup HM) and F M in DP ) with F pd P M = inf n Z semiprojective and Coker n+1) F. projective

5 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 5 The number obtained by replacing semiprojective with semiflat and projective with flat is the flat dimension of M, denoted fd P M. For the rest of this section we fix a factorization K P S of σ, with K P essentially smooth of relative dimension d and P S finite. Lemma 1.4. For every complex M of P -modules the following inequalities hold: fd K M fd P M fd K M + pd P e P. In particular, fd P M and fd K M are finite simultaneously. When the S-module HM) is finite one can replace fd P M with pd P M. Proof. The inequality on the left is a consequence of [3, 4.2F)]. For the one on the right we may assume fd K M = q <. Thus, if F M is a semiflat resolution over P, then G = Coker F q+1) is flat as a K-module. For each n Z there is a canonical isomorphism of functors of P -modules Tor P n G, ) = Tor P e n P, G K ), see [10, X.2.8], so the desired inequality holds. Since K P is essentially smooth one has pd P e P <, see 1.3.4, so they imply that fd P M is finite if only if so is fd K M. In case HM) is finite over P one has fd P M = pd P M; see [3, 2.10F)]. Lemma 1.5. The canonical homomorphisms λ P K d, τ P K d, and ɛ d P K 1.3.1, 1.3.2, and 1.3.3, respectively, provide isomorphisms of P -modules 1.5.1) Ext n P ep, P e ) = 0 for n d ; defined in 1.5.2) Hom S λ P K d, P ) ɛ d P K : Extd P ep, P e ) = Hom P Ω d P K, P ) ; 1.5.3) τ P K d λ P K d : Ω d P K = Hom P Ext d P ep, P e ), P ). Proof. Set I = Kerµ). It suffices to prove that the maps above induce isomorphisms after localization at every n Spec P. Fix one, then set T = P m, R = P e n P e and J = I n P e. The ideal J is generated by a regular sequence, see Any such sequence consists of d elements: This follows from the isomorphisms of T -modules J/J 2 = I/I 2 ) n = Ω 1 P K ) n = T d. The Koszul complex Y on such a sequence is a free resolution of T over R. A well known isomorphism Hom R Y, R) = Σ d Y of complexes of R-modules yields Ext n RT, R) = 0 for n d and Ext d RT, R) = T. This establishes 1.5.1) and shows that Ext d P ep, P e ) is invertible; as a consequence, 1.5.2) follows from 1.5.3). We analyze the maps in 1.5.3). From we know that λ P K d is bijective. By one has τ P K d = κ d H d δ). The map H d δ) is bijective, as it can be computed from a resolution X of P by finite projective P e -modules, and then δ itself is an isomorphism. To establish the isomorphism in 1.5.3) it remains to show that κ d ) m is bijective. This is a Künneth map, which can be computed using the Koszul complex Y above. Thus, we need to show that the natural T -linear map H d Hom R Hom R Y, R), T )) Hom R H d Hom R Y, R)), T ) is bijective. It has been noted above that both modules involved are isomorphic to T, and an easy calculation shows that the map itself is an isomorphism. To continue we need a lemma from general homological algebra.

6 6 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK Lemma 1.6. Let R be an associative ring and M a complex of R-modules. If the graded R-module HM) is projective, then there exists a unique up to homotopy morphism of complexes HM) M inducing id HM), and a unique isomorphism α: HM) M in DR) with Hα) = id HM). Proof. One has HM) = i Z Σi H i M) as complexes with zero differentials. The projectivity of the R-modules H i M) provides the second link in the chain HHom R HM), M)) = H i Z Σ i Hom R H i M), M)) = Σ i Hom R H i M), HM)) i Z = Hom R i Z Σ i H i M), HM)) = Hom R HM), HM)) of isomorphisms of graded modules. The composite map is given by clsα) Hα). The first assertion follows because H 0 Hom R HM), M)) is the set of homotopy classes of morphisms HM) M. For the second, note that one has Mor DR) HM), M) = H 0 Hom R HM), M)) because each complex Σ i H i M) is semiprojective, and hence so is HM). Lemma 1.7. In DP ) there exist canonical isomorphisms 1.7.1) RHom P ep, P e ) Σ d Hom P Ω d P K, P ) ) RHom P RHom P ep, P e ), P ) Σ d Ω d P K. Proof. Since K P is essentially smooth of relative dimension d, the P -module Ω d P K is projective of rank one, and hence so is Hom P Ω d P K, P ). The isomorphisms 1.5.1) and 1.5.2) imply that HRHom P ep, P e )) is an invertible graded P -module. In particular, it is projective. Now choose 1.7.1) to be the canonical isomorphism provided by Lemma 1.6, and 1.7.2) the isomorphism induced by it. Lemma 1.8. When σ is finite there is a canonical isomorphism Σ d RHom P S, Ω d P K ) = RHom K S, K) in DS). Proof. One has a chain of canonical isomorphisms: Σ d RHom P S, Ω d P K ) Σd RHom P ep, RHom K S, Ω d P K )) Σ d RHom P ep, P e ) L P e RHom KS, Ω d P K ) RHom P Ω d P K, P ) L P RHom KS, Ω d e P K ) RHom P Ω d P K, P ) L P Ω d e P K L K RHom K S, K) ) RHom P Ω d P K, P ) L P P L P e Ω d P K L K RHom K S, K) )) RHom P Ω d P K, P ) L P Ω d P K L P RHom K S, K) ) RHom P Ω d P K, Ωd P K ) L P RHom K S, K) RHom K S, K).

7 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 7 The first one holds by a classical associativity formula, see 2.1.1), the second one because pd P e P is finite, see 1.3.4, the third one by 1.7.1). The last one is induced by the homothety P RHom P Ω d P K, Ωd P K ), which is bijective as Ωd P K ) p = P p holds as P p -modules for each p Spec P. The other isomorphisms are standard. Proof of Theorem 1.1. Let K Q S be the factorization of σ given by 1.0.1), with Q = V 1 K[x 1,..., x e ]. The isomorphism Ω 1 P K Q) K = Ω 1 P K K Q) P K Ω 1 Q K ) induces the first isomorphism of P K Q)-modules below: Ω d+e P K Q) K = Ω i P K K Q) P K Q P K Ω j Q K ) i+j=d+e = Ω d P K K Ω e Q K. The second one holds because for each p Spec P one has Ω i P K ) p = i P p P d p ) = 0 for i > d, and similarly Ω i Q K ) p = 0 for j > e. One also has 1.9.1) Ω n P K Q) Q = Ω n P K K Q for every n N. The isomorphisms above explain the first and third links in the chain RHom P K QS, Σ d+e Ω d+e P K Q) K ) RHom P K QS, Σ d Ω d P K K Σ e Ω e Q K ) RHom P K QS, Σ d Ω d P K K Q) Q Σ e Ω e Q K RHom P K QS, Σ d Ω d P K Q) Q ) Q Σ e Ω e Q K RHom Q S, Q) Q Σ e Ω e Q K RHom Q S, Σ e Ω e Q K ) For the fourth isomorphism, apply Lemma 1.8 to the factorization Q P K Q S of the finite homomorphism Q S, where the first map is essentially smooth by [14, v)] and has relative dimension d by 1.9.1). The other isomorphisms are standard. By symmetry one also obtains an isomorphism RHom P K QS, Σ d+e Ω d+e P K Q) K ) RHom P S, Σ d Ω d P K ). Proof of Theorem 1.2. Recall that K P S is a factorization of σ with K P essentially smooth of relative dimension d and P S finite. Set L = Σ d Ω d P K, and note that one has D σ = RHom P S, L); see Theorem ) Standard adjunctions give isomorphisms of functors RHom S, D σ ) = RHom S, RHom P S, L)) = RHom P, L), For M Pσ) Lemma 1.4 yields pd P M <, so M is represented in DP ) by a bounded complex F of finite projective P -modules. As L is a shift of a finite projective P -module, Hom P F, L) is a bounded complex of finite projective P -modules. It represents RHom P M, L), so one sees that HRHom P M, L)) is finite over P. As P acts on it through S, it is finite over S as well; furthermore, fd K RHom P M, L) is finite by Lemma 1.4. The map δ M in DS) is represented in DP ) by the canonical biduality map F Hom P Hom P F, L), L). This is a quasiisomorphism as F is finite complex of finite projectives and L is invertible. It follows that δ M is an isomorphism.

8 8 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK 2) Since fd K S is finite, 1) applied to M = S shows that D σ = RHom S S, D σ ) is in Pσ) and that δ S : S RHom S RHom S S, D σ ), D σ ) is an isomorphism. Composing δ S with the map induced by the isomorphism D σ RHom S S, D σ ) one gets χ Dσ : S RHom S D σ, D σ ), hence χ Dσ is an isomorphism. 2. DG derived categories Our purpose here is to introduce background material on differential graded homological algebra needed to state and prove the results in Sections 3 and 4. In this section K denotes a commutative ring DG algebras and DG modules. Our terminology and conventions generally agree with those of MacLane [22, Ch. VI]. All DG algebras are defined over K, are zero in negative degrees, and act on their DG modules from the left. When A is a DG algebra and N a DG A-module we write A and N for the graded algebra and graded A -module underlying A and N, respectively. We set inf N = inf{n Z N n 0} ; sup N = sup{n Z N n 0}. Every element x N has a well defined degree, denoted x. When B is a DG algebra the complex A K B is a DG algebra with product a b) a b ) = 1) b a aa bb ). When M is a DG B-module the complex N K M is canonically a DG module over A K B, with a b) n m ) = 1) b n an bm. The opposite DG K-algebra A o has the same underlying complex of K-modules as A, and product given by a b = 1) a b ba. We identify right DG A-modules with DG modules over A o, via the formula am = 1) a m ma. When M is a DG B-module the complex Hom K M, N) is canonically a DG A K B o -module, with action given by a b)α) ) m) = 1) b α aαbm). We write A e for the DG K-algebra A K A o. Any morphism α: A B of DG K-algebras induces a morphism α e = α K α o from A e to B e. There is a natural DG A e -module structure on A given by a a )x = 1) a x axa. For every DG A K B o -module L, [22, VI.8.7)] yields a canonical isomorphism 2.1.1) Hom A K B ol, Hom KM, N)) = Hom A L B M, N). For every DG A o K B-module L, [22, VI.8.3)] yields a canonical isomorphism 2.1.2) L A K B o N K M ) = L A N) B M Properties of DG modules. A DG A-module F is said to be semiprojective if the functor Hom A F, ) preserves surjections and quasi-isomorphisms, and semiflat if F A ) preserves injections and quasi-isomorphisms. If F is semiprojective, respectively, semiflat, then F is projective, respectively, flat, over A ; the converse is true when F is bounded below. Semiprojectivity implies semiflatness. A DG module I is semiinjective if Hom A, I) transforms injections into surjections and preserves quasi-isomorphisms. If I is semiinjective, then I is injective over A ; the converse is true when I is bounded above Every quasi-isomorphisms of DG modules, both of which are either semiprojective or semiinjective, is a homotopy equivalence. The following properties readily follow from standard adjunction formulas.

9 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS Let α: A B be a morphism of DG K-algebras, and let X and Y be DG modules over A and B, respectively. The following statements hold: 1) If X is semiprojective, then so is the DG B-module B A X. 2) If X is semiinjective, then so is the DG B-module Hom A B, X). 3) If B is semiprojective over A and Y is semiprojective over B, then Y is semiprojective over A. 4) If B is semiflat over A and Y is semiinjective over B, then Y is semiinjective over A Resolutions of DG modules. Let M be a DG A-module A semiprojective resolution of M is a quasi-isomorphism F M with F semiprojective. Each DG A-module M admits such a resolution; [4, 1]. A semiinjective resolution of M is a quasi-isomorphism M I with I semiinjective. Every DG A-module M admits such a resolution; see [18, 3-2]. In what follows, for each DG module M over A, we fix a semiprojective resolution πa M : p AM) M, and a semiinjective resolution ι M A : M i AM). Each morphism of DG modules lifts up to homotopy to a morphism of their semiprojective resolutions and extends to a morphism of their semiinjective resolutions, and such a lifting or extension is unique up to homotopy. In particular, both F and I are unique up to homotopy equivalences inducing the identity on M. Lemma Let ω : A B be a quasi-isomorphism of DG algebras, I a semiinjective DG A-module, J a semiinjective DG B-module, and ι: J I a quasiisomorphism of DG A-modules. For every DG B-module L the following map is a quasi-isomorphism: Hom ω L, ι): Hom B L, J) Hom A L, I). Proof. The morphism ι factors as a composition J ι Hom A B, I) Hom Aω,I) Hom A A, I) = I of morphisms of DG A-modules, where ι x)b) = 1) x b bιx). It follows that ι is a quasi-isomorphism. Now J is a semiinjective DG B-module by hypothesis, Hom A B, J) is one by ), so yields Hom B L, J) Hom B L, Hom A B, I)) = Hom A L, I). It remains to note that the composition of these maps is equal to Hom ω L, ι). Lemma Let ω : A B be a morphism of DG algebras, and let Y and Y be DG B-modules that are quasi-isomorphic when viewed as DG A-modules. If ω is a quasi-isomorphism, or if there exists a morphism β : B A, such that ωβ = id B, then Y and Y are quasi-isomorphic as DG B-modules. Proof. By hypothesis, one has A-linear quasi-isomorphisms Y υ U υ Y. When ω is a quasi-isomorphism, choose U semiprojective over A, using With vertical arrows defined to be b u bυu) and b u bυ u) the diagram U υ A A U υ ω AU Y B A U Y

10 10 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK commutes. The vertical maps are morphisms of DG B-modules, and ω A U is a quasi-isomorphism because ω is one and U is semiprojective. When ω has a right inverse β, note that the A-linear quasi-isomorphisms υ and υ are also B-linear, and that the DG B-module structures on Y and Y induced via β are identical with their original structures over B. We recall basic facts concerning DG derived categories; see Keller [18] for details DG derived categories. Let A be a DG algebra and M a DG A-module. DG A-modules and their morphisms form an abelian category. The derived category DA) is obtained by keeping the same objects and by formally inverting all quasi-isomorphisms. It has a natural triangulation, with translation functor Σ is defined on M by ΣM) i = M i 1, ΣM ςm) = ς M m)), and aςm) = 1) a ςam), where ς : M ΣM is the degree one map given by ςm) = m. For any semiprojective resolution F M, and each N DA) one has Mor DR) M, N) = H 0 Hom R F, N)) For all L DA o ) and M, N in DA), the complexes of K-modules L L A M = L A F and RHom A M, N) = Hom A F, N) are defined uniquely up to unique isomorphisms in DA). When ω : A B is a morphism of DG algebras, L, M and N are DG B-modules, and λ: L L, µ: M M, and ν : N N are ω-equivariant morphisms of DG modules, there exist uniquely defined morphisms λ L ω µ: L L A M L L B M, RHom ω µ, ν): RHom B M, N ) RHom A M, N). that depend functorially on all three arguments, and are isomorphisms when all the morphisms involved have this property. For each i Z one sets Tor A i L, M) = H i L L A M) and Ext i AM, N) = H i RHom A M, N)) Associative K-algebras are viewed as DG algebras concentrated in degree zero, in which case DG modules are simply complexes of left modules. Graded modules are complexes with zero differential, and modules are complexes concentrated in degree zero. The constructions above specialize to familiar concepts: When A i = 0 for i 0 the derived category DA) coincides with the classical unbounded derived category of the category of A 0 -modules. Similarly, if M and N are DG A-modules with M i = 0 = N i for i 0, then for all n Z one has Ext n AM, N) = Ext n A 0 M 0, N 0 ) and Tor A n M, N) = Tor A0 n M 0, N 0 ) Let ω : A B be a morphism of DG algebras. Viewing DG B-modules as DG A-modules via restriction along ω, one gets a functor of derived categories ω : DB) DA). When ω is a quasi-isomorphism it is an equivalence, with quasi-inverse B L A. 3. Derived Hochschild functors In this section we explain the left hand side of the isomorphism in Theorem 1. Let K be a commutative ring and σ : K S an associative K-algebra.

11 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS A flat DG algebra resolution of σ is a factorization K A α S of σ as a composition of morphisms of DG algebras, where each K-module A i is flat and α is a quasi-isomorphism; complexes of S-modules are viewed as DG A-modules via α. When K B β S is a flat DG algebra resolution of σ, we say that ω : A B is a morphism of resolutions if it is a morphism of DG K-algebras, satisfying βω = α. We set A e = A K A o, note that K A o αo S o is a flat DG algebra resolution of σ o : K S o, and turn S into a DG module over A e by a a )s = αa)s α o a ). Flat DG algebra resolutions always exist: A resolution K T S, with T the tensor algebra of some free non-negatively graded K-module, can be obtained by inductively adjoining noncommuting variables to K; see also Lemma 3.7. Here we construct one of four functors of pairs of complexes of S-modules that can be obtained by combining RHom A es, ) and S L A with e L K ) and RHom K, ). The other three functors are briefly discussed in 3.10 and The statement of the following theorem is related to results in [32, 2]. We provide a detailed proof, for reasons explained in Theorem 3.2. Each flat DG algebra resolution K A S of σ defines a functor RHom A es, L K ): DS) DS o ) DS c ), where S c denote the center of S, described by 3.8.1). For every flat DG algebra resolution K B S of σ there is a canonical natural equivalence of functors ω AB : RHom A es, L K ) RHom B es, L K ), given by 3.8.2), and every flat DG algebra resolution K C S of σ satisfies ω AC = ω BC ω AB. The theorem validates the following notation: Remark 3.3. Fix a flat DG algebra resolution K A S of σ and let RHom S L K S os, L K ): DS) DS o ) DS c ) denote the functor RHom A K A os, L K ). For all L DS) and L DS o ) it yields derived Hochschild cohomology modules with tensor-decomposable coefficients: Ext n S L K So S, L L K L ) = H n RHom S L K S os, L L K L )). These modules are related to vintage Hochschild cohomology. For all S-modules L and L there are canonical natural maps HH n S K; L K L ) Ext n S K S os, L K L ) of S c -modules, where the modules on the left are the classical ones, see These are isomorphisms when S is K-projective; see [10, IX, 6]. When one of L or L is K-flat, there exist canonical natural homomorphisms Ext n α K α os, L K L ): Ext n S K S os, L K L ) Ext n S L K So S, L K L ). When S is K-flat the composition K S = S is a flat DG resolution of σ and α: A S is a morphism of resolutions, so the theorem shows that the maps above are isomorphisms.

12 12 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK Construction 3.4. Let K A α S and K A α S o be flat DG algebra resolutions of σ and of σ o, respectively. We turn S into a DG module over A K A by setting a a )s = αa)s α a ). The action of S c on S commutes with that of A K A, and so confers a natural structure of complex of S c -modules on Hom A K A S, i A K A p AL) K p A L )), where p A and i A K A refer to the resolutions introduced in Let K B β S and K B β S o be DG algebra resolutions of σ and σ o, respectively, and ω : A B and ω : A B be morphism of resolutions. We turn DG B-modules into DG A-modules via ω, and remark that the equality βω = α implies that on S-modules the new action of A coincides with the old one. Let λ: L M be a morphism of DG S-modules and λ : L M one of DG S o -modules. The lifting property of semiprojective DG modules yields diagrams 3.4.1) p A L) L eλ λ p B M) M and p A L ) eλ p B M ) L λ M of DG A-modules and DG A -modules, respectively, that commute up to homotopy. It provides the morphism in the top row of a diagram of DG A K A )-modules p A L) K p A L ) eλ Kλ e p B M) K p B M ) 3.4.2) i A K A p AL) K p A L )) ɛ i A K A ib K B p BM) K p B M )) ) ι i B K B p BM) K p B M )) that commutes up to homotopy, where ι is the chosen semiinjective resolution, and ɛ is given by the extension property of semiinjective DG module over A K A ; for conciseness, we rewrite these maps as E ɛ I ι J. They are unique up to homotopy, as the liftings and extensions used for their construction have this property. The hypotheses βω = α and β ω = α imply that ω and ω are quasi-isomorphisms, hence so is ω K ω, due to the K-flatness of A and B. Since ι is a quasi-isomorphism, Lemma shows that so is Hom ω K ω S, ι); thus, the latter map defines in DS c ) an isomorphism, denoted RHom ω K ω S, ι). We set 3.4.3) [ω, ω ]λ, λ ) = RHom ω K ω S, ι) 1 RHom A K A S, ɛ): RHom A K A S, L L K L ) RHom B K B S, M L K M ) The first statement of the following lemma contains the existence of the functors RHom A es, L K ), asserted in the theorem. The second statement, concerning the uniqueness of these functors, is weaker than the desired one, because it only applies to resolutions that can be compared through a morphism ω : A B. On the other hand, it allows one to compare functors defined by independently chosen resolutions of σ and σ o. The extra generality is needed in the proof of Lemma 3.7.

13 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 13 Lemma 3.5. In the notation of Construction 3.4, the assignment defines a functor and the assignment L, L ) Hom A K A S, i A K A p AL) K p A L ))), RHom A K A S, L K ): DS) DS o ) DS c ), λ, λ ) [ω, ω ]λ, λ ), given by formula 3.4.3), defines a canonical natural equivalence of functors [ω, ω ]: RHom A K A S, L K ) RHom B K B S, L K ). If K C γ S and K C γ S are flat DG algebra resolutions of σ and σ o, respectively, and ϑ: B C and ϑ : B C are morphism of resolutions, then [ϑω, ϑ ω ] = [ϑ, ϑ ][ω, ω ]. Proof. Recall that the maps E ɛ ι I J are unique up to homotopy. Thus, Hom A K A S, ɛ) and Hom ω K ω S, ι) are morphisms of complexes of Sc -modules defined uniquely up to homotopy. In view of 3.4.3), this uniqueness has the following consequences: The morphism [ω, ω ]λ, λ ) depends only on λ and λ ; one has [id A, id A ]id L, id L ) = id RHom A K A S,L L K L ) ; and for all morphism µ: M N and µ : M N of complexes of S-modules and S o -modules, respectively, there are equalities [ϑω, ϑ ω ]µλ, µ λ ) = [ϑ, ϑ ]µ, µ ) [ω, ω ]λ, λ ). Suitable specializations of these properties show that RHom A K A S, L K ) is a functor to DS c ) from the product of the categories of complexes over S with that of complexes over S o, and that [ω, ω ] is a natural transformation. To prove that [ω, ω ] is an equivalence, it suffices to show that if λ and λ are quasi-isomorphisms, then RHom ω K ω S, λ L K λ ) is an isomorphism. By 3.4.3), it is enough to show that RHom A K A S, ɛ) is a quasi-isomorphism. As λ and λ are quasi-isomorphisms, the diagrams in 3.4.1) imply that so are λ and λ. Due to the K-flatness of A and B, their semiprojective DG modules are K-flat, hence λ λ K is a quasi-isomorphism of DG modules over A K A. Now diagram 3.4.2) shows that ɛ: E I is a quasi-isomorphism. It follows that it is a homotopy equivalence, because both E and J are semiinjective DG modules over A K A. This implies that Hom A K A S, ɛ) is a quasi-isomorphism, as desired. To clarify how the natural equivalence in Lemmas 3.5 depends on ω, we apply Quillen s homotopical approach in [25]. It is made available by the following result, see Baues and Pirashvili [8, A.3.1, A.3.5]: 3.6. The category of DG K-algebras has a model structure, where the weak equivalences are the quasi-isomorphisms; the fibrations are the morphisms that are surjective in positive degrees; any DG K-algebra, whose underlying graded algebra is the tensor algebra of a non-negatively graded projective K-module, is cofibrant; that is, the structure map from K is a cofibration.

14 14 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK We recall some consequences of the existence of a model structure, following [12]: For all DG K-algebras T and A, there exists a relation on the set of morphisms T A, known as left homotopy, see [12, 4.2]. It is an equivalence when T is cofibrant, see [12, 4.7], and then π l T, A) denotes the set of equivalence classes. Lemma 3.7. There is a DG algebra resolution K T S of σ with T cofibrant. If K A α S is a flat DG algebra resolutions of σ, then there is a morphism of resolutions ω : T A. Any morphism of resolutions ϖ : T A is left homotopic to ω, and the natural equivalences defined in Lemma 3.5 satisfy [ω, ω o ] = [ϖ, ϖ o ]: RHom T es, L K ) RHom A es, L K ) Proof. Being both a fibration and a weak equivalence, α is, by definition, an acyclic fibration. The existence of ω comes from a defining property of model categories the left lifting property of cofibrations with respect to acyclic fibrations; see axiom MC4i) in [12, 3.3]. Composition with α induces a bijection π l T, A) π l T, S), see [12, 4.9], so αϖ = αω implies that ϖ and ω are left homotopic. By [12, 4.3, 4.4], the homotopy relation produces a commutative diagram T ρ T C T of DG K-algebras, with a quasi-isomorphism ρ. It induces a commutative diagram ι ι χ ω ϖ A T K T o ω K T ι K T o o T K T o ρ C K T o χ K T o K T o A K T o ι K T o T K T o ϖ K T o of morphisms of DG K-algebras, where ρ K T o is a quasi-isomorphism because T o is K-flat. The diagram above yields the following chain of equalities: [ω, id T o ] = [χ, id T o ][ι, id T o ] = [χ, id T o ][ρ, id T o ] 1 = [χ, id T o ][ι, id T o ] = [ϖ, id T o ]. A similar argument shows that the morphisms ω o and ϖ o are left homotopic, and yields [id A, ω o ] = [id A, ϖ o ]. Assembling these data, one obtains [ω, ω o ] = [id A, ω o ][ω, id T o ] = [id A, ϖ o ][ϖ, id T o ] = [ϖ, ϖ o ]. Proof of Theorem 3.2. Choose a DG algebra resolution K T S of σ with T cofibrant, either by noting that the one in 3.1 has this property by 3.6, or referring to a defining property of model categories; see axiom MC5i) in [12, 3.3]. For each flat DG algebra resolution K A S of σ, form the flat DG algebra resolution K A o S o of σ o, and define a functor 3.8.1) RHom A es, L K ): DS) DS o ) DS c )

15 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 15 by applying Lemma 3.5 with A = A o. As T is cofibrant, Lemma 3.7 provides a morphism of resolutions ω : T A, and shows that it defines a natural equivalence [ω, ω o ]: RHom T es, L K ) RHom A es, L K ) ; that does not depend on the choice of ω; set ω A T = [ω, ωo ]. When K U S also is a flat DG algebra resolution of σ with U cofibrant, one gets morphisms of resolutions τ : T U and θ : U A. Both θτ : T A and ω are morphisms of resolutions, so Lemmas 3.7 and 3.5 yield ω A T = [ω, ω o ] = [θτ, θ o τ o ] = [θ, θ o ][τ, τ o ] = ω A U ω U T. For each flat DG algebra resolution K B S of σ set 3.8.2) ω AB := ω B T ω A T ) 1 : RHom A es, L K ) RHom B es, L K ). One clearly has ω AC = ω BC ω AB, and ω AB is independent of T, because ω B T ω A T ) 1 = ω B U ω U T ω A U ω U T ) 1 = ω B U ω U T ω U T ) 1 ω A U ) 1 = ω B U ω A U ) 1. It follows that ω AB is the desired canonical natural equivalence. We proceed with a short discussion of other derived Hochschild functors. The proof of the next result is omitted, as it parallels that of Theorem 3.2. Theorem 3.9. Any flat DG algebra resolution K A S of σ defines a functor A A e RHom K, ): DS) op DS) DS c ). For each flat DG algebra resolution K B S of σ one has a canonical equivalence ω BA : B B e RHom K, ) A A e RHom K, ) of functors, and every flat DG algebra resolution K C S of σ satisfies ω CA = ω BA ω CB. Remark We fix a DG algebra resolution K A S of σ and let S L S L K So RHom K, ): DS) op DS) DS c ) denote the functor A A K A RHom K, ): The preceding theorem shows that it is independent of the choice of A. For all M, N DS) it defines derived Hochschild homology modules of the K-algebra S with Hom-decomposable coefficients: Tor S L K So n S, RHom K M, N)) = H n S L S L K So RHom K M, N)). These modules are related to classical Hochschild homology: For all S-modules M and N there are canonical natural maps Tor S KS o n S, Hom K M, N)) HH n S K; Hom K M, N)) of S c -modules, where the modules on the left are the classical ones, see They are isomorphisms when S is K-flat; see [10, IX, 6]. When M is K-projective there exist natural homomorphisms Tor α Kα o n S, RHom K M, N)): Tor S L K So n S, RHom K M, N)) Tor S KS o n S, Hom K M, N)) When S is K-flat the composition K S = S is a flat DG resolution of σ and α: A S is a morphism of resolutions, so the theorem shows that the maps above are isomorphisms.

16 16 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK The remaining two composed functors collapse in a predictable way. Remark Similarly to Theorems 3.2 and 3.9, one can define functors RHom S L K S os, RHom K, )): DS) op DS) DS c ), S S L K S o L K ): DS) DS) DS c ), that do not depend on the choice of the DG algebra resolution A. However, this is not necessary, as for all M, N DS) there exit canonical isomorphisms ) ) RHom S L K S os, RHom KM, N)) RHom S M, N), S S L K S o M L K N) M L S N. They are derived extensions of classical reduction results, [10, IX.2.8, IX.2.8a]. We finish with a comparison of the content of this section and that of [32, 2]. Remark When M = N the statement of Theorem 3.2 bears a close resemblance to results of Yekutieli and Zhang, see [32, 2.2, 2.3]. One might ask whether their proof can be adapted to handle the general case. Unfortunately, even in the special case above the argument for [32, Theorem 2.2] is deficient. It utilizes the mapping cylinder of morphisms φ 0, φ 1 : M M of DG modules over a DG algebra, B. On page 3225, line 11, they are described as the two B -linear quasi-isomorphisms φ 0 and φ 1 where B is a DG algebra equipped with two homomorphisms of DG algebras u 0, u 1 : B B; with this, an implicit choice is being made between u 0 and u 1. Such a choice compromises the argument, whose goal is to establish an equality χ 0 = χ 1 between morphism of complexes χ i, which have already been constructed by using φ i and u i for i = 0, 1. The basic problem is that the relation between various choices of comparison morphisms of DG algebra resolutions is not registered in the additive environment of derived categories. In the proof of Theorem 3.2 it is solved by using the homotopy equivalence provided by a model structure on the category of DG algebras. 4. Reduction of derived Hochschild functors over algebras Let σ : K S be a homomorphism of commutative rings. Recall that σ is said to be essentially of finite type if it can be factored as K K[x 1,..., x d ] V 1 K[x 1,..., x d ] S, where x 1,..., x d are indeterminates, V is a multiplicatively closed subset, the first two maps are canonical, and the third one is a surjective ring homomorphism. The following theorem, which is the main algebraic result in the paper, involves the relative dualizing complex D σ described in 1.0.2). Theorem 4.1. If fd K S is finite, then in DS) there are isomorphisms 4.1.1) 4.1.2) RHom S L K SS, M L K N) RHom S RHom S M, D σ ), N) RHom S L K SS, RHom S M, D σ ) L K N) RHom S M, N) for all M Pσ) and N DS); these morphisms are natural in M and N. We record a useful special case, obtained by combining Theorems 4.1 and 1.1:

17 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 17 Corollary 4.2. Assume that σ is flat, and let K P S be a factorization of σ with K P essentially smooth of relative dimension d and P S finite. If M is a finite S-module that is flat over K, and N is an S-module, then for each n Z there is an isomorphism of S-modules Ext n S K SS, M K N) = Ext n d S RHom P M, Ω d P K ), N). Before the proof of Theorem 4.1 we make a couple of remarks For all complexes of P -modules L, X, and J there is a natural morphism Hom P L, P ) P X P J Hom P Hom P X, L), J) defined by the assignment λ x j χ 1) x + j ) λ λχx)j ). This morphism is bijective when L and X are finite projective: This is clear when L and X are shifts of P. The case when they are shifts of projective modules follows, as the functors involved commute with finite direct sums. The general case is obtained by induction on the number of the degrees in which L and X are not zero A DG algebra A is called graded-commutative if ab = 1) a b ba holds for all a, b A. The identity map A o A then is a morphism of DG algebras, so each DG A-module is canonically a DG module over A o, and for all A-modules M and N the complexes RHom A M, N) and M L A N are canonically DG A-modules. When A and B are graded-commutative DG algebras, then so is A K B, and the canonical isomorphisms in 2.1.1) and 2.1.2) represent morphisms in DA K B). Proof of Theorem 4.1. The argument proceeds in several steps, with notation introduced as needed. It uses chains of quasi-isomorphisms that involve a number of auxiliary DG algebras and DG modules. We start with the DG algebras. Step 1. There exists a commutative diagram of morphisms of DG K-algebras P e η e B e σ K S κ π µ P β ι η η e P e P α A B e η P e P B e P e A = B ν e P e α µ B e P e A C C B P B B e P e ι where flags quasi-isomorphisms and tips surjections. The morphisms appearing in the diagram are constructed in the following sequence: Fix a factorization K κ P π S of σ, with κ essentially smooth of relative dimension d and π finite. Set P e = P K P and let µ: P e P denote the multiplication map, and note that the projective dimension pd P e P is finite by γ

18 18 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK Choose a graded-commutative DG algebra resolution P e ι α A P of µ with A 0 = P e, each A i a finite projective P e -module, and sup A = pd P e P ; see [2, 2.2.8]. Choose a graded-commutative DG algebra resolution P η B π S of σ, with B 0 a finite free P -module and each B i a finite free P -module; again, see [2, 2.2.8]. Set B e = B K B and let µ : B P B B be the multiplication map. Let ν : B e P e P B P B be the map b b p b b )p. Let γ : B e P e A B P B be the map b b a b b )αa). The diagram commutes by construction. The map ν is an isomorphism by 2.1.2), and B e P e α is a quasi-isomorphism because α is one and B e is a bounded below complex of flat P e -modules. We always specify the DG algebra operating on any newly introduced DG module. On DG modules of homomorphisms and tensor products the operations are those induced from the arguments of these functors; see 4.4. Notation. Let P e U be a semiinjective resolution over P e. Set H = HHom P ep, U)). Step 2. There exists a unique isomorphism inducing id H in homology: H RHom P ep, P e ) in DP ). Proof. The isomorphism H = Ext P ep, P e ) of graded P -modules and 1.7.1) show that H is projective, so Lemma 1.6 applies. Notation. Set L = Hom P H, P ). Let L I be a semiinjective resolution over P. Step 3. There exists an isomorphism D σ RHom P S, I) in DS). Proof. Theorem 1.1 provides the first isomorphism in the chain D σ RHom P S, Σ d Ω d P K ) RHom P S, RHom P RHom P ep, P e ), P )) RHom P S, RHom P H, P )) RHom P S, I). The remaining ones come from 1.7.2), Step 2, and the resolution L I. Notation. Let X M be a semiprojective resolution over B, with X i a finite projective P -module for each i and inf X = inf HM), see 2.3.1; set q = pd P M, and note that q is finite by Lemma 1.4. Set X = X /X, where X i = X i for i > q, X q = X q+1 ), and X i = 0 for i < q. It is easy to see that X is a DG submodule of X, so the canonical map X X is a surjective quasi-isomorphism of DG B-modules. Since X M is a semiprojective resolution over P, each P -module X i is projective; see [3, 2.4.P]. Let G Hom P X, L) be a semiprojective resolution over B. Let N J be a semiinjective resolution over B. Set J = Hom B S, J). Step 4. There exists an isomorphism RHom S RHom S M, D σ ), N) RHom B RHom P M, L), N) in DB).

19 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 19 Proof. The map N J induces the vertical arrows in the commutative diagram N Hom B S, N) J Hom B S, J) = Hom B β,n) Hom B β,j) Hom B B, N) Hom B B, J) Note that B acts on N through β, which is surjective, so Hom B β, N) is bijective. The map Hom B β, J) is a quasi-isomorphism because β is one and J is semiinjective. By ), J is semiinjective, so N J is a semiinjective resolution over S. In the following chain of morphisms of DG B-modules the isomorphisms are adjunctions: Hom S Hom S M, Hom P S, I)), J) = Hom S Hom P M, I), J) = Hom S Hom P M, I), Hom B S, J)) = Hom B S S Hom P M, I), J) = Hom B Hom P M, I), J) Hom B Hom P X, I), J) Hom B Hom P X, I), J) Hom B Hom P X, L), J) Hom B G, J) The quasi-isomorphisms are induced by M X X, L I, and G X, because I is semiinjective over P, J is semiinjective over B, and X is semiprojective over P. The chain yields the desired isomorphism in DB) as J is semiinjective over S, G is semiprojective over B, and Step 3 gives Hom P S, I) D σ. Notation. Let F B be a semiprojective resolution over C. Step 5. There exists an isomorphism RHom B RHom P M, L), N) RHom C B, RHom P RHom P M, L), N)) in DC). Proof. The DG C-module C C F is semiprojective by ). The map F B induces the vertical arrows in the commutative diagram of DG C-modules F B = C C F = C C B γ C F = γ C B C C F C C B where γ C B is an isomorphism because γ is surjective and C acts on B through γ, and γ C F is a quasi-isomorphism because γ is one and F is semiprojective. The resulting quasi-isomorphism C C F B induces the quasi-isomorphism in the following chain, because Hom P G, J) is semiinjective over C by ): Hom B G, J) = Hom CB, Hom P G, J)) Hom C C C F, Hom P G, J)) = Hom C F, Hom P G, J)).

20 20 L. L. AVRAMOV, S. B. IYENGAR, J. LIPMAN, AND S. NAYAK The first isomorphism reflects the action of C = B P B on Hom P G, J), the second one holds by adjunction. The chain represents the desired isomorphism because Hom P G, J) is semiinjective over C; see 4.4. Notation. Let Y N be a semiprojective resolution over B. Step 6. There exists an isomorphism RHom P RHom P M, L), N) RHom P ep, P e ) L P e M L K N) in DB e ). Proof. From G Hom P X, L) one gets the first link in the chain Hom P G, J) Hom P Hom P X, L), J) = Hom P L, P ) P X P J = H P X P J H P X P Y = H P e X K Y ) Hom P ep, U) P e X K Y ) of morphisms of DG C-modules; it is a quasi-isomorphism because the semiinjective DG B-module J is semiinjective over P, see ). The equality reflects the definition of L. The composition Y N J induces the third link; which is a quasiisomorphism because H and X are semiprojective over P. The second isomorphism holds by associativity of tensor products; see The quasi-isomorphism H Hom P ea, P e ) from Step 2 induces the last link, which is a quasi-isomorphism because X K Y is semiflat over P e. Finally, the semiinjectivity of J and the semiflatness of X K Y imply that the chain above represents the desired isomorphism in DB e ). Step 7. There exists an isomorphism RHom P ep, P e ) L P e M L K N) RHom P ep, M L K N) in DB e ). Proof. The resolutions A P U over P e induce quasi-isomorphisms Hom P ep, U) Hom P ea, U) Hom P ea, P e ) of complexes of P e -modules, which in turn induce a quasi-isomorphism Hom P ep, U) P e X K Y ) Hom P ea, P e ) P e X K Y ) of DG B e -modules. To wrap things up, we use the canonical evaluation morphism Hom P ea, P e ) P e X K Y ) Hom P ea, X K Y ) given by λ x y a 1) x + y ) a λa)x y) ) ; it is bijective, because the DG algebra A is a bounded complex of finite projective P e -modules. Notation. Let X K Y V be a semiinjective resolution over B e. Step 8. There exists an isomorphism RHom C B, RHom P ep, M L K N)) RHom B es, M L K N) in DB e ).

21 REDUCTION OF DERIVED HOCHSCHILD FUNCTORS 21 Proof. The isomorphisms below come from adjunction formulas, see 2.1.1): Hom C F, Hom P ea, X K Y )) = Hom B ef A A, X K Y ) = Hom B ef, X K Y ) Hom B ef, V ) Hom B es, V ) The quasi-isomorphisms are induced by X K Y V and F S, respectively, because F is semiprojective over B e and V is semiinjective over B e. Step 9. The composed morphism of the chain of isomorphisms RHom S RHom S M, D σ ), N) RHom B RHom P M, L), N) RHom C B, RHom P RHom P M, L), N)) RHom C B, RHom P ep, P e ) L P e M L K N)) RHom C B, RHom P ep, M L K N)) RHom B es, M L K N) RHom S L K SS, M L K N) provided by Steps 4 through 8 and Theorem 3.2, defines an isomorphism in DS). Proof. The diagram of DG algebras in Step 1 provides a morphism from B e to every DG algebra appearing in the chain of canonical isomorphisms above. Thus, each isomorphism in the chain above defines a unique isomorphism in DB e ). Its source and target are complexes of S-modules, on which B e acts through the composed morphism of DG algebras B e B S. This map is equal to the composition B e S e S. Therefore, Lemma 2.3.3, applied first to the quasi-isomorphism B e S e, then to the homomorphisms S S e S given by s s 1 and s s ss, shows that the complexes above are also isomorphic in DS). Step 10. The morphism in Step 9 is natural with respect to M and N. Proof. The morphism in question is represented by a composition of quasi-isomorphisms of DG modules over B e, so it suffices to verify that each such quasi-isomorphism represents a natural morphism in DB e ). Three kinds of quasi-isomorphisms are used. The one chosen in Step 2 involves neither M nor N, and so works simultaneously for all complexes of S-modules; thus, no issues of naturality arises there. Some of the constituent quasi-isomorphisms themselves are natural isomorphisms, such as Hom-tensor adjunction or associativity of tensor products. Finally, there are quasi-isomorphisms of functors induced replacing some DG module with a semiprojective or a semiinjective resolution. The induced morphism of derived functors are natural, because morphisms of DG modules define unique up to homotopy morphisms of their resolutions; see The isomorphism 4.1.1) and its properties have now been established. Theorem 1.21) shows that formula 4.1.2) is equivalent to 4.1.1). The next result is an analog of Theorem 4.1 for the derived Hochschild functor from Remark 3.10; it can be proved along the same lines, so the argument is omitted.